Book of Proof - Amazon S3

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is even,” or “Every even integer greater than 2 is the sum of two primes.”
(The latter statement is called the Goldbach conjecture. See Section 2.1.)
Mathematicians have a special name for the statements in this category
that they suspect (but haven’t yet proved) are true. Such statements are
called conjectures.
Three Types of Statements:
Known to be true Truth unknown Known to be false
(Theorems & propositions)
(Conjectures)
Examples:
• Pythagorean theorem
• Fermat’s last theorem
(Section 2.1)
• The square of an odd
number is odd.
∞∑ 1
• The series
k=1 k diverges.
Examples:
• All perfect numbers are
even.
• Any even number greater
than 2 is the sum of two
primes. (Goldbach’s
conjecture, Section 2.1)
• There are infinitely many
prime numbers of form
2 n − 1, with n ∈ N.
Examples:
• All prime numbers are
odd.
• Some quadratic equations
have three solutions.
• 0 = 1
• There exist natural
numbers a, b and c
for which a 3 + b 3 = c 3 .
Mathematicians spend much of their time and energy attempting
to prove or disprove conjectures. (They also expend considerable mental
energy in creating new conjectures based on collected evidence or intuition.)
When a conjecture is proved (or disproved) the proof or disproof will
typically appear in a published paper, provided the conjecture is of sufficient
interest. If it is proved, the conjecture attains the status of a theorem or
proposition. If it is disproved, then no one is really very interested in it
anymore—mathematicians do not care much for false statements.
Most conjectures that mathematicians are interested in are quite
difficult to prove or disprove. We are not at that level yet. In this text, the
“conjectures” that you will encounter are the kinds of statements that an
experienced mathematician would immediately spot as true or false, but
you may have to do some work before figuring out a proof or disproof. But
in keeping with the cloud of uncertainty that surrounds conjectures at the
advanced levels of mathematics, most exercises in this chapter (and many
beyond it) will ask you to prove or disprove statements without giving any
hint as to whether they are true or false. Your job will be to decide whether
or not they are true and to either prove or disprove them. The examples
in this chapter will illustrate the processes one typically goes through in